Lattice of closure endomorphisms of a Hilbert algebra
Abstract
A closure endomorphism of a Hilbert algebra A is a mapping that is simultaneously an endomorphism of and a closure operator on A. It is known that the set CE of all closure endomorphisms of A is a distributive lattice where the meet of two elements is defined pointwise and their join is given by their composition. This lattice is shown in the paper to be isomorphic to the lattice of certain filters of A, antiisomorphic to the lattice of certain closure retracts of A, and compactly generated. The set of compact elements of CE coincides with the adjoint semilattice of A, conditions under which two Hilbert algebras have isomorphic adjoint semilattices (equivalently, minimal Brouwerian extensions) are discussed. Several consequences are drawn also for implication algebras.
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 DOI:
 10.48550/arXiv.1701.03902
 arXiv:
 arXiv:1701.03902
 Bibcode:
 2017arXiv170103902C
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Logic;
 03G25;
 06A15;
 06D99;
 06F35;
 08A35
 EPrint:
 16 pages, no figures, submitted to Algebra Universalis (under review since 24.11.2015)